The Coulomb field around an electric charge Q should express phenomena which are analogous to the metric of the Schwarzschild solution around a mass M.
Though we are not seeing this phenomena in the spectrum of the hydrogen atom that is put to a high or low electric potential. A possible explanation is that the extra inertia in an electric field has a delay: energy must actually flow in the common electric field of Q and q for the inertia to become visible. Since the electron orbits the proton at a very high frequency, the inertia does not have time to show up. Or there might be a quantum mechanical reason why an electron in a stationary state does not show extra inertia.
Macroscopic charges should show extra inertia since the Poynting vector tells us that energy flows in the common field of q and Q.
Increased inertia slows down clocks and contracts rulers in the radial direction close to a mass M
A clock ticks slower.
● M
m • \/\/\/\/\/\/\ spring
<------>
oscillation
The spacetime itself has not changed in any way. A mechanical clock based on a spring harmonic oscillator,
F = m a = k x,
oscillates slower because the inertia of m has grown by a factor
1 / sqrt(1 - rₛ / r),
and the elastic energy stored in the spring has been diluted by a factor
sqrt(1 - rₛ / r),
since that elastic energy is in a low gravity potential. The effective spring constant k is reduced by the same factor. The oscillation angular speed is
ω = sqrt(k / m).
Close to M, the spring constant is reduced and the inertia of m is larger, so that the oscillation speed is reduced by a factor
sqrt(1 - rₛ / r).
A ruler is shorter when turned to the radial direction. The inertia of a test mass m is larger in the radial direction than in the tangential direction relative to M. This is because when m moves closer to M, energy is shipped to m from the common field of m and M, and that energy is shipped, on the average, over a distance r.
Since the inertia is larger to one direction, all physical processes are contracted in that direction. The atoms in the ruler are squeezed in the radial direction. That is why the ruler is shortened by a factor
sqrt(1 - r_s / r)
in the radial direction. Light propagates slower in the radial direction.
The precession of the perihelion of Mercury is due to these changes in inertia.
Clocks tick slower and rulers are shortened around a charge Q?
Close to a positive charge Q, a negative test charge experiences changes in its inertia, similar to a test mass m close to M.
Let us assume that both Q and q are 1 microcoulomb, which is a large static electric charge in everyday life. The distance is 1 meter. The extra inertia is
1 / c² * 1 / (4 π ε₀) * 10⁻⁶ C * 10⁻⁶ C / 1 m
= 10⁻¹⁹ kg.
It is hardly measurable in a laboratory.
The spring of the oscillator does not grow weaker in an electric field, because elastic energy is electrically neutral. Thus, the "metric of time" near a charge Q only slows down for certain clocks, and less because energy is not diluted close to Q.
The inertia of an electron moving radially relative to the charge Q should be larger than for an electron moving tangentially. But since we are not seeing the effect inertia in the spectrum of hydrogen, we do not think that rulers really contract in the radial direction relative to Q.
How does an electric potential affect the propagation of electromagnetic waves?
In our September 4, 2023 blog post we claimed that an electromagnetic wave must be based on hidden electric currents in "empty" space. It is analogous to an electromagnetic wave propagating inside a material. An electric field increases the inertia of charges inside a material, and must slow down the propagation of waves somewhat.
How can we calculate the effect in "empty" space?
● Q
s = distance (Q, cube)
----------
| | ^ E
| | | ~~~~~~~~
---------- <-- wave
s
The procedure might be the same as calculating the effect inside a material. To create an electric field E inside a cubic volume s³ in space, we have to displace a charge q per the area s² for a distance s. The electric field is then
E = 1 / ε₀ * q / s²
<=>
q = ε₀ E s².
Suppose that the beam of the wave is s wide and s high. For the wave to be able to pass through the cube, its wavelength must be < s. Let us assume that it is 2 s. If the wavelength is shorter, we can make the cube smaller.
When the wave propagates a distance s, it must ship the charge q over the distance s. The associated energy shipping is of the order
W' = q E' s
= ε₀ E E' s³
over a distance s. There E' is the electric field of the charge Q.
The energy of the wave in that volume of the cube s³ is
W = s³ * ε₀ E²,
where E is the maximum electric field in the wave.
The inertia of the wave increases roughly by a factor
1 + W' / W
= 2,
if E' = E. The result is in a stark contradiction with empirical results! An electric field has no measured effect on the propagation of electromagnetic waves in vacuum.
The reason might be that the field E' of Q makes the alternating field E to carry more energy because the energy density really is
ε₀ (E + E')²,
when the external field E' is present. The wave E can reduce its amplitude, compensating entirely the extra inertia which was put on the shoulders of the wave. Then the inertia of the wave stays constant as it passes through the external field E'. For the wave to transport its energy through the field E', it is enough that the combined field E + E' transports the energy.
Another explanation: the hidden electric currents may contain an equal amount of positive and negative charges. In our September 4, 2023 blog post about magnetism, we claim that close to a positive charge +Q, the inertia of a negative test charge -q grows, but the inertia of a positive +q test charge is reduced. The effects would cancel each other.
Gravitational waves propagate without distortion or delay in the field of a mass M? Probably no
Our observation of electromagnetic waves in an external field Q has an interesting ramification for gravitational waves. Do they pass through a gravity field of a mass M unhindered, with no slowing down or refraction?
Traditionally, people think that M modifies the "geometry" of spacetime, and gravitational waves must obey the geometry and be refracted.
In general relativity, gravitational waves do not contain "explicit energy". The stress-energy tensor is zero. This suggests that they might propagate along paths which are prohibited from ordinary mass-energy.
Question. Does the gravity field of a mass affect the propagation of a gravitational wave in any way?
If the field equations for gravity are linear, then we can superpose a static gravity field and a wave, and it is still a solution of the field equations. The wave propagates unaffected by the static field.
If we assume that all energy gravitates, then a pulse of gravitational waves must interact with the field of M and slow down. This introduces nonlinearity to the field equations of the gravity field. Thus, the argument about linear field equations does not hold.
We conclude that gravitational waves are almost certainly affected by the field of M.
Does an electric potential affect the propagation of an electromagnetic wave inside a material?
Inside a material, an electromagnetic wave makes electrons to oscillate. If we could increase the inertia of the electrons, then, presumably, a pulse of light would pass the material slower?
The John Kerr effect makes any material birefringent, that is, changes the refractive index if light is polarized in the direction of the applied electric field E.
However, increasing the inertia of electrons should affect the refractive index in every direction?
To resolve this matter, we should understand the origin of the refractive index of a material. A photon acquires inertia from the material and propagates slower than c, but how does this happen?
Maybe there is a quantum mechanical phenomenon which makes the charge carriers in most materials almost balanced between electrons and "holes" which have a positive charge? Then an electric potential would have a small or an nonexistent effect on the propagation of light in the material.
Let us calculate the refractive index of a material, assuming that harmonic oscillators, each containing one electron, carry the wave. Do we get a reasonable result?
We assume that the materia is solid hydrogen, where the radius of an atom is 0.05 nanometers. Each electron acts as a spring harmonic oscillator where the spring constant k is such that
1 eV = 1/2 k * (0.01 nm)²
<=>
k = 2 * 10²² eV/m²
= 3,200 N/m.
Let the maximum electric field in the wave be E. The displacement
E e = k x
<=>
x = E e / k
and the energy
W = 1/2 k x²
= 1/2 k E² e² / k²
= 1/2 e² E² / k
= E² * 4 * 10⁻⁴² Jm²/V².
Let us have one cubic meter of the material. The energy of the wave in it is
W' = ε₀ * E²
= E² * 10⁻¹¹ Jm²/V².
The energy of the oscillators in it is
E² * (10¹⁰)³ * 4 * 10⁻⁴²
= E² * 4 * 10⁻¹² Jm²/V².
When a one cubic meter wave packet enters one cubic meter of the material, maybe 40% of its energy is absorbed by the oscillators for some time. The oscillators eventually give back all the energy and momentum to the wave packet as it leaves the material.
A hypothesis: while an oscillator is holding the energy W, the momentum which was possessed by that energy W pushes the oscillator forward. The material keeps the momentum for some time. As the oscillator radiates the energy W back to the wave, it returns the momentum back to the wave.
The speed of the wave is then only 0.6 c, which corresponds to a refractive index 1 / 0.6 = 1.7.
The value is reasonable.
If we put an electron to a potential of 50 kV, caused by a positive charge Q, its inertia should increase by 10%, and the refractive index should change considerably. But we do not see the effect. Possible explanations:
1. There is a delay in getting the inertia. If the inertia comes from the common field of Q and the electron, maybe centimeters away, it does not have time to affect a wave whose wavelength is ~ 1 micrometer.
2. The oscillator is in a semistationary quantum state, and the electric field of the oscillator is zero farther than 0.1 nanometers away (or farther than ~ micrometer away). There is no extra inertia because Q does not know of the oscillation at all. The charge Q does not know about the orbit of the electron in a regular hydrogen atom, either.
3. Charge carriers are a mix of electrons and "holes". If the inertia of the electrons increases, that is balanced by the decrease of the inertia of "holes".
Conclusions
Electrons in a microscopic context do not seem to express changes of inertia inside an external electric field. The reason may be quantum mechanical.
For macroscopic charges, there should be changes in the inertia, if we believe the Poynting vector. However, static macroscopic charges in a laboratory are very small relative to the mass of the object which is carrying the charge. Measuring the inertia is probably impossible.
It may be possible to measure the inertia of a free electron under a potential. We checked one such measurement about two years ago, but the test setting contained conducting materials which would screen any change in the inertia.
No comments:
Post a Comment